The number of points of intersection of curve $\sin x = \cos y$ and circle $x^2 + y^2 = 1$.
Step-by-Step Solution
Key Concept: The range of $\cos\theta \pm \sin\theta$ is bounded by $\sqrt{2}$, which cannot equal $\frac{\pi}{2} - 2n\pi$ for any integer $n$.
A point $(\cos\theta, \sin\theta)$ on the circle lies on the curve $\sin x = \cos y$ when $\sin(\cos\theta) = \cos(\sin\theta)$. This leads to the condition $\cos\theta \pm \sin\theta = \frac{\pi}{2} - 2n\pi$. Since $|\cos\theta \pm \sin\theta| \leq \sqrt{2}$ for all $\theta$, and $\frac{\pi}{2} - 2n\pi > \sqrt{2}$ for all integers $n$, no solution exists.
Correct Answer: 0